For charter flights things along those lines already exist. For example hopping a ride on empty leg repositioning routes.

For truly private jet flights it can be trickier. Many truly private jets operate as private flight operations under part 91 of the FAA’s rules. That has a lot less red tape for a private operator but means they truly need to be a private operator and can’t pickup others for hire.

Several startups have tried to get around the above via legal loopholes via ‘Uber for airplanes’ models and got shot down by the FAA.